Hamiltonian Dynamical Systems

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

Modern Physics Letters B ◽

10.1142/s0217984989001813 ◽

1989 ◽

Vol 03 (15) ◽

pp. 1185-1188 ◽

Cited By ~ 1

Author(s):

J. SEIMENIS

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Infinite Number ◽

Chaotic Behavior ◽

Equations Of Motion ◽

Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

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On the Analysis of Time-Periodic Nonlinear Hamiltonian Dynamical Systems

Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems ◽

10.1115/detc1995-0277 ◽

1995 ◽

Author(s):

Eric A. Butcher ◽

S. C. Sinha

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Normal Form ◽

Linear Part ◽

Normal Form Theory ◽

Mathieu’S Equation ◽

Form Theory ◽

Hamiltonian Dynamical Systems ◽

Time Invariant ◽

Time Periodic

Abstract In this paper, some analysis techniques for general time-periodic nonlinear Hamiltonian dynamical systems have been presented. Unlike the traditional perturbation or averaging methods, these techniques are applicable to systems whose Hamiltonians contain ‘strong’ parametric excitation terms. First, the well-known Liapunov-Floquet (L-F) transformation is utilized to convert the time-periodic dynamical system to a form in which the linear pan is time invariant. At this stage two viable alternatives are suggested. In the first approach, the resulting dynamical system is transformed to a Hamiltonian normal form through an application of permutation matrices. It is demonstrated that this approach is simple and straightforward as opposed to the traditional methods where a complicated set of algebraic manipulations are required. Since these operations yield Hamiltonians whose quadratic parts are integrable and time-invariant, further analysis can be carried out by the application of action-angle coordinate transformation and Hamiltonian perturbation theory. In the second approach, the resulting quasilinear time-periodic system (with a time-invariant linear part) is directly analyzed via time-dependent normal form theory. In many instances, the system can be analyzed via time-independent normal form theory or by the method of averaging. Examples of a nonlinear Mathieu’s equation and coupled nonlinear Mathieu’s equations are included and some preliminary results are presented.

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